On computing an eigenvector of a tridiagonal matrix. Part I: basic results

SIAM Journal on Matrix Analysis and Applications

Volumn 18, Issue 4, 1997, Pages 1013-1034

  Fernando, K V ^a  

^a Numerical Algorithms Group   (United Kingdom)

Author keywords

Deflation; Eigenvalues; Eigenvectors; Inverse iteration; Perturbation analysis; Tridiagonal matrices

Indexed keywords

EID: 0031536846     PISSN: 08954798     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0895479895294484     Document Type: Article

References (19)

1. E. ANDERSON, Z. BAI, C. BISCHOF, J. DEMMEL, J. DONGARRA, J. DU CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, S. OSTROUCHOV, AND D. SORENSEN, LAPACK Users' Guide, Release 2.0, SIAM, Philadelphia, 1995.
2. I. BABUŠKA, Numerical stability in problems of linear algebra, SIAM J. Numer. Anal., 9 (1972), pp. 53-77.
3. R. E. D. BISHOP, G. M. L. GLADWELL, AND S. MICHAELSON, Matrix Analysis of Vibrations, Cambridge University Press, Cambridge, 1965.
4. e Série). Vol. 9, Gauthier-Villars, Paris, 1841.
5. J. DEMMEL AND W. KAHAN, Accurate singular values of bidiagonal matrices, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 873-912.
6. K. V. FERNANDO, Accurate BABE Factorisation of Tridiagonal Matrices for Eigenproblems, Technical Report TR5/95, Numerical Algorithms Group Ltd, Wilkinson House, Jordan Hill, Oxford, 1995.
7. K. V. FERNANDO, On a Classical Method for Computing Eigenvectors, Technical Report TR3/95, Numerical Algorithms Group Ltd, Wilkinson House, Jordan Hill, Oxford, 1995.
8. K. V. FERNANDO, Computing an Eigenvector of a Tridiagonal When the Eigenvector is Known, in Proc. of the Third International Congress on Industrial and Applied Mathematics, Zeitschrift für Angewandte Mathematik und Mechanik, Hamburg, 1996.
9. G. H. GOLUB AND W. KAHAN, Calculating the singular values and pseudo-inverse of a matrix, SIAM J. Numer. Anal., 2 (1965), pp. 205-224.
10. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, The Johns Hopkins University Press, Baltimore, MD, 1989.
11. P. HENRICI, Bounds for eigenvalues of certain tridiagonal matrices, J. Soc. Indust. Appl. Math., 1 (1963), pp. 281-290.
12. H. HOLZER, Die Berechnung der Drehschwingungen und ihre Anwendung im Maschinenbau, Verlag von Julius Springer, Berlin, 1921.
13. E. R. JESSUP AND I. C. F. IPSEN, Improving the accuracy of inverse iteration, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 550-571.
14. W. KAHAN, Accurate Eigenvalues of a Symmetric Tri-Diagonal Matrix, Technical Report CS 41, Computer Science Department, Stanford University, Stanford, CA, 1966.
15. J. SHERMAN AND W. J. MORRISON, Adjustment of an inverse matrix corresponding to a change in one element of a given matrix, Ann. Math. Statist., 21 (1950), pp. 124-126.
16. H. S. WALL, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948.
17. J. H. WILKINSON, The calculation of eigenvectors of codiagonal matrices, Computer J., 1 (1958), pp. 90-96.
18. J. H. WILKINSON, Rounding Errors in Algebraic Processes, Her Majesty's Stationary Office, London, 1963.
19. J. H. WILKINSON, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965.